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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 940 · 40 questions
TalentJR

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

NameDateTime takenScore ___ / 40
  1. 1.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 31 (c) 11 (d) 29
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
  3. 3.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 5476 (d) 4884
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  5. 5.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
  6. 6.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6734 (d) 6404
  7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
  8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  9. 9.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  10. 10.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 171 (c) 117 (d) 225
  11. 11.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 149 (c) 429 (d) 289
  12. 12.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 9 (d) 11
  13. 13.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
  14. 14.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
  15. 15.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 32 (c) 33 (d) 12
  16. 16.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 4
  17. 17.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 12 (c) 26 (d) 27
  18. 18.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 6416 (c) 5776 (d) 7066
  20. 20.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 484 (d) 396
  22. 22.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 49 (d) 14
  23. 23.What are the roots of x² − 11x + 18 = 0? (a) 1 and 10 (b) 3 and 6 (c) 2 and 9 (d) 1 and 18
  24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6724 (c) 6396 (d) 6094
  26. 26.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 256 (d) 130
  27. 27.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 289 (c) 7 (d) 49
  28. 28.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2116 (c) 2516 (d) 2916
  29. 29.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 6375 (c) 7225 (d) 5635
  30. 30.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 337 (c) 169 (d) 225
  31. 31.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
  32. 32.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 151 (d) 106
  33. 33.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3969 (d) 3249
  34. 34.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
  35. 35.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 16 (c) 4 (d) 144
  36. 36.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2209 (d) 2809
  37. 37.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9025 (d) 9975
  38. 38.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  39. 39.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 19 (c) 21 (d) 9
  40. 40.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3844 (d) 3604

Answer key

  1. 30
  2. 17
  3. 4356
  4. (x + 8)(x + 9)
  5. 4
  6. 6724
  7. 72
  8. (x + 6)(x + 7)
  9. (x + 3)(x + 5)
  10. 117
  11. 149
  12. 10
  13. 8 and 9
  14. 16
  15. 32
  16. 5
  17. 27
  18. (x + 4)(x + 7)
  19. 7056
  20. (x + 2)(x + 4)
  21. 324
  22. 48
  23. 2 and 9
  24. (x + 3)(x + 6)
  25. 6084
  26. 130
  27. 49
  28. 2916
  29. 5625
  30. 113
  31. (x + 2)(x + 7)
  32. 106
  33. 3969
  34. (x + 2)(x + 6)
  35. 16
  36. 2809
  37. 9025
  38. (x + 7)(x + 8)
  39. 20
  40. 3844

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